So, the Universe started with a bang. Everything was hot, dense, and expanding.

It’s 13.7 billion years later now, and our Universe is cold and sparse. The temperature of the leftover glow from the big bang — which used to be over 10^30 degrees — is now down to 2.7 Kelvin, just barely above absolute zero. The Universe used to be denser than the center of the Sun. Now, on average, the density of the Universe is only about one proton per cubic meter, with mass clumped into stars and galaxies separated by trillions of miles.

But, for all of it, the Universe is still expanding. What’s amazing is, despite all of the things that could cause this expansion, there’s only one thing that the expansion rate depends on. Know what it is?

The energy density of the Universe. Or, in other words, the total amount of matter, radiation, neutrinos, and all other forms of energy in the entire Universe divided by the Universe’s volume. I’ll say it again: energy density is the only thing that determines the rate of expansion of the Universe.

This is remarkable, and here’s why. First off, the Universe could have existed in one of three ways. It could have been curved like a sphere (what we call positive), curved like a saddle (what we call negative), or completely flat (what we call zero curvature).

If the Universe were either positively or negatively curved, the three angles of triangles in space would not add up to 180 degrees! And the expansion rate would be dramatically changed by the presence of curvature. The fact that we have none is still a mystery for us, although perhaps inflation explains why.

The fact that the expansion rate depends on energy density means something else: at different times in the Universe, different things were important. Think about this: if you have a Universe where you double the volume, the matter density halves. Right? Same mass, twice the volume, therefore the density is half. I mean, look, the coins on the bigger ballon are clearly less dense, right? Same number of coins, but they’re spread out over a larger volume:

But radiation — things like light — not only dilute like matter does, it gets redshifted, which means its energy density drops faster than matter does. So when the Universe was very young and very hot and very dense, radiation was more important than matter was!

As the Universe cooled, matter became more important, and dominated the expansion rate for billions of years. But there’s something else in the Universe, something that doesn’t dilute as space expands. It appears that there’s an energy inherent in space itself, and this is what we call dark energy. Some people call it a cosmological constant. Why constant? Because its energy density never drops. Just a few billion years ago, dark energy became the dominant form of energy in the Universe. Matter gets less dense, but dark energy never does. If we graphed them all together — radiation, matter, and dark energy — you’d see which one dominated the expansion rate of the Universe.

And that’s it. That’s all that causes the Universe’s expansion: the total amount of energy that’s in it. From here to infinity, as far as we can tell, dark energy will continue to dominate the Universe.

Some of you are inclined towards math, and will want an equation that relates the Hubble expansion rate (H) to the energy density (ρ). Well, I hate getting too technical, so here you go:

Want the more detailed equations? Go here, if you can’t wait. I’ll do a special, simple math post tomorrow for those of you who can.

Slightly off-topic question: I am looking for a rough estimate for the number of black holes in the universe as a function of the scale factor, and how that number would develop into the future if expansion would decelerate, cease, accelerate. Ideally also their distribution over mass. This doesn’t have to be very sophisticated. Do you have any advice or reference? Best,

Nick, part of the maddening thing about dark energy is that there are no known particles that have this property. The only thing we know of that behaves this way (watch out for technical words) are three-dimensional topological defects, known as “textures”. Our Universe doesn’t appear to have them, and so we invent a new type of field instead of a particle, and call it dark energy.

Nicholas, you’re welcome! Friday, I promise.

Bee, hi! I’ve liked your blog, backreaction, for a long time. The difficulty with what you’re asking is the assumption that black hole formation is at all related to the expansion of the Universe. While this is true in the very early Universe (when it’s all radiation dominated anyway), once you head on over into the later stages, all black hole formation occurs inside of virialized galaxies. I throw “virialized” in there to emphasize that the evolution of individual galaxies is not dependent on the expansion rate of the Universe. The only way you’d get an effect is if black holes form during galaxy mergers, in which case you could calculate the future merger rate under differing cosmologies. This is probably your best bet, although I’d be searching papers the same as you after this point to find the answer.

Thanks for the kind words. Well, I haven’t said the expansion of the universe needs to make a difference. I was just thinking it would. If there’s a recollapse, I would naively have thought stuff will clump increasingly. In any case, as long as black hole formation continues, at some point they either clump among each other or just sit there. I’d think the number would reach an asymptotic value if a(t) continues to grow, but not sure what if it has a max and bounces.

If you come across a useful reference, would you let me know? (Preferably by email, I might not check back here till the end of time, it’s sabine at perimeterinstitute dot ca). As I said, it doesn’t have to be very accurate, some handwavy estimate would be fine as long as somehow reasonable.

“…. Or, in other words, the total amount of matter, radiation, neutrinos, and all other forms of energy in the entire Universe divided by the Universe’s volume.”

Pray tell me, how can you measure and put a finite value to the total volume of the Universe? I have recently visited the Hubble website and was gob-smacked by the sheer scale of the universe out there. Some photos have light dots which represent galaxies billions of time bigger than ours. Also Feynman in a recorded interview made and interesting remark about the physical world we are aware of, through the pin-holes of our eyes. An ant on a carpet in a living room could not extrapolate the total size of the carpet, let alone what house it was in. Hubble and Chandra are only mapping a tiny part of the Universe. So, how can you put a value on the total size of the Universe?

Thanks, Didac. I am a retired layman so please forgive the goofy questions.
Your reply assumes that the density is constant throughout the universe. Is this an assumption or evidence based?
By the way this blog is fantastic. Good work.

what part of the univers is expanding?
If you look at the total univers all distance are growing and its growing faster and faster. No not true, some of the distance are growing but not everyone. That is weird.
If some of it is growing and some of it is not, what is then holding the distance steady in some part of the univers. Gravity, gravity is keeping elements of the univers to gether. Gravity and the expanding univers is going in opposite directions, this must create heat or dark energy. The string theory makes this look weird because some part of the string is expandin and some is not. Is it possible that the expanding univers is trying to expand our solarsystem but the gravity is keeping it steady.
Is it possible that the expanding univers is trying to expand the sun but the gravity is keeping it toegether, and this is making the sun “burning”, when the energy of the sun is going down it will expand and die. Is it possible that the expanding univers i draging the earth to earthquake and vulcanos?.

Interessting idea, this reminds me of the physics class that we had in school and the detailed discussions about the Universe and if it will ever stop to expand. I recently started to get back into this topic and like it a lot. Thanks for the explanations. sebastian mode from Vienna Austria